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# 7.1 The HMM Parameters

A HMM consists of a number of states. Each state j has an associated observation probability distribution which determines the probability of generating observation at time t and each pair of states i and j has an associated transition probability . In HTK the entry state 1 and the exit state N of an N state HMM are non-emitting.

Fig. 7.1 shows a simple left-right HMM with five states in total. Three of these are emitting states and have output probability distributions associated with them. The transition matrix for this model will have 5 rows and 5 columns. Each row will sum to one except for the final row which is always all zero since no transitions are allowed out of the final state.

HTK is principally concerned with continuous  density models in which each observation probability distribution is represented by a mixture Gaussian density. In this case, for state j the probability of generating observation is given by

where is the number of mixture components  in state j for stream s, is the weight of the m'th component and is a multivariate Gaussian with mean vector  and covariance matrix  , that is

where n is the dimensionality of . The exponent is a stream weight  and its default value is one. Other values can be used to emphasise particular streams, however, none of the standard HTK tools manipulate it.

HTK also supports discrete probability distributions  in which case

where is the output of the vector quantiser for stream s given input vector and is the probability of state j generating symbol v in stream s.

In addition to the above, any model or state can have an associated vector of duration parameters   . Also, it is necessary to specify the kind of the observation vectors, and the width of the observation vector in each stream. Thus, the total information needed to define a single HMM is as follows

• type of observation vector
• number and width of each data stream
• optional model duration parameter vector
• number of states
• for each emitting state and each stream
• mixture component weights or discrete probabilities
• if continuous density, then means and covariances
• optional stream weight vector
• optional duration parameter vector
• transition matrix
The following sections explain how these are defined.

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